toe

Acronym toe

TOCID symbol tO, tTT

Name truncated octahedron,
Voronoi cell of body-centered cubic (bcc) lattice

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Circumradius sqrt(5/2) = 1.581139

Vertex figure [4,6²] = qo&#h

Snub derivation

Vertex layers

Layer	Symmetry	Subsymmetries
	`o3o4o`	`o3o .`	`o . o`	`. o4o`
1	`x3x4o`	`x3x .` {6} first	`x . o` edge first	`. x4o` {4} first
2		`u3x .`	`u . q`	`. u4o`
3a		`x3u .`	`w . o`	`. x4q`
3b		`x3u .`	`x . Q`	`. x4q`
4a		`x3x .` opposite {6}	`w . o`	`. u4o`
4b		`x3x .` opposite {6}	`x . Q`	`. u4o`
5			`u . q`	`. x4o` opposite {4}
6			`x . o`
	`o3o3o`	`o3o .`	`o . o`	`. o3o`
1	`x3x3x`	`x3x .` {6} first	`x . x` {4} first	`. x3x` {6} first
2		`x3u .`	`u . u`	`. u3x`
3a		`u3x .`	`x . w`	`. x3u`
3b		`u3x .`	`w . x`	`. x3u`
4		`x3x .` opposite {6}	`u . u`	`. x3x` opposite {6}
5			`x . x`

Lace city
in approx. ASCII-art

  o q o  
o   Q   o   (Q=2q)
q Q   Q q
o   Q   o
  o q o

      x      
   u     u   
x     w     x
             
x     w     x
   u     u   
      x

Coordinates (sqrt(2), 1/sqrt(2), 0) & all permutations, all changes of sign

General of army (is itself convex)

Colonel of regiment (is itself locally convex – no other uniform polyhedral members)

Confer

variations:: a3b3c x3x3u a3b4c x3f4o x3u4o x3w4o u3x4o

External
links

Incidence matrix according to Dynkin symbol

x3x4o

. . . | 24 |  1  2 | 2 1
------+----+-------+----
x . . |  2 | 12  * | 2 0
. x . |  2 |  * 24 | 1 1
------+----+-------+----
x3x . |  6 |  3  3 | 8 *
. x4o |  4 |  0  4 | * 6

x3x4/3o

. .   . | 24 |  1  2 | 2 1
--------+----+-------+----
x .   . |  2 | 12  * | 2 0
. x   . |  2 |  * 24 | 1 1
--------+----+-------+----
x3x   . |  6 |  3  3 | 8 *
. x4/3o |  4 |  0  4 | * 6

x3x3x

. . . | 24 |  1  1  1 | 1 1 1
------+----+----------+------
x . . |  2 | 12  *  * | 1 1 0
. x . |  2 |  * 12  * | 1 0 1
. . x |  2 |  *  * 12 | 0 1 1
------+----+----------+------
x3x . |  6 |  3  3  0 | 4 * *
x . x |  4 |  2  0  2 | * 6 *
. x3x |  6 |  0  3  3 | * * 4

s4x3x

demi( . . . ) | 24 |  1  1  1 | 1 1 1
--------------+----+----------+------
demi( . x . ) |  2 | 12  *  * | 1 1 0
demi( . . x ) |  2 |  * 12  * | 0 1 1
sefa( s4x . ) |  2 |  *  * 12 | 1 0 1
--------------+----+----------+------
      s4x .   ♦  4 |  2  0  2 | 6 * *
demi( . x3x ) |  6 |  3  3  0 | * 4 *
sefa( s4x3x ) |  6 |  0  3  3 | * * 4

xuxux4ooqoo&#xt   → all heights = 1/sqrt(2) = 0.707107
({4} || pseudo u-{4} || pseudo (x,q)-{8} || pseudo u-{4} || {4})

o....4o....     | 4 * * * * | 2 1 0 0 0 0 0 | 1 2 0 0 0
.o...4.o...     | * 4 * * * | 0 1 2 0 0 0 0 | 0 2 1 0 0
..o..4..o..     | * * 8 * * | 0 0 1 1 1 0 0 | 0 1 1 1 0
...o.4...o.     | * * * 4 * | 0 0 0 0 2 1 0 | 0 0 1 2 0
....o4....o     | * * * * 4 | 0 0 0 0 0 1 2 | 0 0 0 2 1
----------------+-----------+---------------+----------
x.... .....     | 2 0 0 0 0 | 4 * * * * * * | 1 1 0 0 0
oo...4oo...&#x  | 1 1 0 0 0 | * 4 * * * * * | 0 2 0 0 0
.oo..4.oo..&#x  | 0 1 1 0 0 | * * 8 * * * * | 0 1 1 0 0
..x.. .....     | 0 0 2 0 0 | * * * 4 * * * | 0 1 0 1 0
..oo.4..oo.&#x  | 0 0 1 1 0 | * * * * 8 * * | 0 0 1 1 0
...oo4...oo&#x  | 0 0 0 1 1 | * * * * * 4 * | 0 0 0 2 0
....x .....     | 0 0 0 0 2 | * * * * * * 4 | 0 0 0 1 1
----------------+-----------+---------------+----------
x....4o....     | 4 0 0 0 0 | 4 0 0 0 0 0 0 | 1 * * * *
xux.. .....&#xt | 2 2 2 0 0 | 1 2 2 1 0 0 0 | * 4 * * *
..... .oqo.&#xt | 0 1 2 1 0 | 0 0 2 0 2 0 0 | * * 4 * *
..xux .....&#xt | 0 0 2 2 2 | 0 0 0 1 2 2 1 | * * * 4 *
....x4....o     | 0 0 0 0 4 | 0 0 0 0 0 0 4 | * * * * 1

or
o....4o....      & | 8 * * | 2 1  0 0 | 1 2 0
.o...4.o...      & | * 8 * | 0 1  2 0 | 0 2 1
..o..4..o..        | * * 8 | 0 0  2 1 | 0 2 1
-------------------+-------+----------+------
x.... .....      & | 2 0 0 | 8 *  * * | 1 1 0
oo...4oo...&#x   & | 1 1 0 | * 8  * * | 0 2 0
.oo..4.oo..&#x   & | 0 1 1 | * * 16 * | 0 1 1
..x.. .....        | 0 0 2 | * *  * 4 | 0 2 0
-------------------+-------+----------+------
x....4o....      & | 4 0 0 | 4 0  0 0 | 2 * *
xux.. .....&#xt  & | 2 2 2 | 1 2  2 1 | * 8 *
..... .oqo.&#xt    | 0 2 2 | 0 0  4 0 | * * 4

xxux3xuxx&#xt   → all heights = sqrt(2/3) = 0.816497
({6} || pseudo (x,u)-{6} || pseudo (u,x)-{6} || {6})

o...3o...     | 6 * * * | 1 1 1 0 0 0 0 0 0 | 1 1 1 0 0 0
.o..3.o..     | * 6 * * | 0 0 1 1 1 0 0 0 0 | 0 1 1 1 0 0
..o.3..o.     | * * 6 * | 0 0 0 0 1 1 1 0 0 | 0 0 1 1 1 0
...o3...o     | * * * 6 | 0 0 0 0 0 0 1 1 1 | 0 0 0 1 1 1
--------------+---------+-------------------+------------
x... ....     | 2 0 0 0 | 3 * * * * * * * * | 1 1 0 0 0 0
.... x...     | 2 0 0 0 | * 3 * * * * * * * | 1 0 1 0 0 0
oo..3oo..&#x  | 1 1 0 0 | * * 6 * * * * * * | 0 1 1 0 0 0
.x.. ....     | 0 2 0 0 | * * * 3 * * * * * | 0 1 0 1 0 0
.oo.3.oo.&#x  | 0 1 1 0 | * * * * 6 * * * * | 0 0 1 1 0 0
.... ..x.     | 0 0 2 0 | * * * * * 3 * * * | 0 0 1 0 1 0
..oo3..oo&#x  | 0 0 1 1 | * * * * * * 6 * * | 0 0 0 1 1 0
...x ....     | 0 0 0 2 | * * * * * * * 3 * | 0 0 0 1 0 1
.... ...x     | 0 0 0 2 | * * * * * * * * 3 | 0 0 0 0 1 1
--------------+---------+-------------------+------------
x...3x...     | 6 0 0 0 | 3 3 0 0 0 0 0 0 0 | 1 * * * * *
xx.. ....&#x  | 2 2 0 0 | 1 0 2 1 0 0 0 0 0 | * 3 * * * *
.... xux.&#xt | 2 2 2 0 | 0 1 2 0 2 1 0 0 0 | * * 3 * * *
.xux ....&#xt | 0 2 2 2 | 0 0 0 1 2 0 2 1 0 | * * * 3 * *
.... ..xx&#x  | 0 0 2 2 | 0 0 0 0 0 1 2 0 1 | * * * * 3 *
...x3...x     | 0 0 0 6 | 0 0 0 0 0 0 0 3 3 | * * * * * 1

or
o...3o...      & | 12  * | 1 1  1 0 0 | 1 1 1
.o..3.o..      & |  * 12 | 0 0  1 1 1 | 0 1 2
-----------------+-------+------------+------
x... ....      & |  2  0 | 6 *  * * * | 1 1 0
.... x...      & |  2  0 | * 6  * * * | 1 0 1
oo..3oo..&#x   & |  1  1 | * * 12 * * | 0 1 1
.x.. ....      & |  0  2 | * *  * 6 * | 0 1 1
.oo.3.oo.&#x     |  0  2 | * *  * * 6 | 0 0 2
-----------------+-------+------------+------
x...3x...      & |  6  0 | 3 3  0 0 0 | 2 * *
xx.. ....&#x   & |  2  2 | 1 0  2 1 0 | * 6 *
.... xux.&#xt  & |  2  4 | 0 1  2 1 2 | * * 6

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